“I think it is now or the Prime Minister will lead the Conservatives into the next election,” he told reporters at a Westminster news conference.

“You find MPs privately who will say to you they think that is a really good idea in any number and I would be quite surprised.”

His warning came as Downing Street confirmed Mrs May will meet European Commission president Jean-Claude Juncker for talks in Brussels on Wednesday ahead of a special EU Brexit summit on Sunday.

In a move likely to be welcomed by Brexiters, Number 10 also confirmed that the Government would look at potential technological solutions to keep the Irish border open.

Mrs May’s official spokesman confirmed that references in last week’s draft agreement to “alternative arrangements” for the border could involve technical measures of the kind previously promoted as the “maximum facilitation” solution, or Max Fac.

Last week, senior ERG figures were confidently predicting they would get the 48 letters of no confidence from MPs needed to trigger a vote in Mrs May’s leadership.

But with Brexiteers apparently divided over whether it was the right time to mount a challenge, the prospect of them achieving their target appeared to be dwindling.

(PA Graphics)

However, Mr Rees-Mogg warned that they needed to consider whether they really wanted to carry on with her at the helm.

While party rules would permit a fresh challenge in a year’s time if there was a failed attempt now, he said that in practice it was doubtful that would happen.

“Basically, if there is a vote of confidence it is not just for a year,” he said.

“Getting the 48 letters has shown to be quite difficult, so the idea that in a year you just repeat the process and then she would go at that point, I don’t think that is realistic.”

Asked if his attempt to unseat the Prime Minister had been exposed as a “Dad’s Army” operation, Mr Rees-Mogg replied: “I’ve always admired Captain Mainwaring.”

Theresa May will meet European Commission president Jean-Claude Juncker in Brussels on Wednesday (Stefan Rousseau/PA)

Senior ministers sought to rally round Mrs May, with Justice Secretary David Gauke warning that any move to oust her would be “dangerous for the country”.

He told BBC Breakfast: “The idea that at this point, in the middle of a very delicate negotiation, that is hugely important to the future of this country, that we should remove the Prime Minister, essentially leave us leaderless, for certainly several weeks, possibly months, would be hugely irresponsible.

“I don’t think people should be talking about removing her at this point. I think that would be self-indulgent and dangerous for the country.”

The latest threat to Mrs May followed a shot across the bows from the Democratic Unionist Party, who prop up her minority Government in the Commons.

On Monday night, the party joined Labour in voting against the Government on a Budget measure while abstaining in two other divisions in apparent breach of their “confidence and supply” agreement.

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The party’s deputy leader Nigel Dodds made clear the DUP was ready to continue voting against the Government and called on Mrs May to ditch her proposals and “work for a better deal”.

Mr Dodds said that the agreement with Conservatives committed the DUP to pursue the shared objectives of strengthening the Union and seeking a Brexit that benefits all parts of the UK.

While the DUP had “kept to our word” on the agreement, the resignation of several Conservative ministers showed that Mrs May’s deal “does not represent those shared objectives”, he said.

He added: “The Government will require DUP support to deliver its domestic agenda. We will continue to use our influence for the good of everyone across the United Kingdom.

“If the Government can look beyond a Withdrawal Agreement, which is uniting people from across the political spectrum against it, and instead work towards a better deal, then an outcome can be delivered that truly works to benefit all parts of the United Kingdom.”

Both Brexiteer Boris Johnson and Chancellor Philip Hammond are due to attend the DUP’s conference at the weekend, according to Co Fermanagh newspaper the Impartial Reporter.

In another cause for concern for the PM, Mrs May’s Spanish counterpart Pedro Sanchez said Madrid will vote against the withdrawal deal at the summit on Sunday if Gibraltar’s future is not considered a bilateral issue between the UK and Spain.

In a bid to sway opinion, Mrs May has insisted that her draft Brexit deal puts Northern Ireland in a “fantastic position” for the future.

In an opinion piece published in the Belfast Telegraph, the Prime Minister claimed the region’s constitutional status within the UK had been guaranteed in the agreement.

Mrs May acknowledged that there had been a lot of focus on the Irish border “backstop” and said she “understood and share some of the concerns that have been expressed”.

But she said the backstop was an “acceptable insurance policy” due to provisions in the deal.